# Kevin Buzzard's MSRI Summer School on automorphic forms.

Hi!

## MSRI Summer School

Featuring me, Rebecca Bellovin, Jackie Lang and Ila Varma, trying to teach some graduate students about the Langlands program. Here's the schedule.

## Goals of the summer school.

The main idea is to give people an introduction to the theory of automorphic forms, and in particular to my understanding of what Langlands' conjectures are.

More precisely, I want to introduce and give examples of the things which occur when describing the local and global Langlands conjectures. So I will talk about smooth irreducible representations of p-adic groups, local and global Galois groups and their representations, and automorphic forms and representations. I will also say a little about the theory at infinity (so (g,K)-modules and so on).

I will spend a little time going through the classical definitions of modular forms, and then I will spend more time explaining how to construct the automorphic representation associated to a classical cuspidal modular form which is an eigenform for the Hecke operators.

If time permits I might even spend some time on the trace formula, and how it can be used to prove simple cases of Langlands' functoriality.

## Random things which might help.

I learnt the basics of the theory (with a strong emphasis on GL(2)) from a course which Richard Taylor gave in CalTech in 1992. Not particularly good scans of my handwritten (in blue pen :-/ ) notes are here.

I also learnt a lot from the instructional programme run at the Newton Institute on this sort of thing, in early 1993. Scans of my notes (this time written in black ink) are here

I found the book "Algebraic number theory" by Cassels-Froehlich very helpful when I was learning stuff. Also, the conference proceedings commonly referred to as "Corvallis" (Proc Sympos Pure Math Volume 33, parts 1 and 2) was also very helpful; initially I used to look at Tate's article a lot, and I got older I started looking more carefully at the other articles.

I wrote a paper with Toby Gee called "The conjectural connections between automorphic representations and Galois representations". I suppose one could argue that the MSRI summer school is my attempt to teach the background necessary for reading that paper The paper is here.

Over the years I've written various notes on things like automorphic forms for GL(2) over Q, automorphic forms in general, Maass forms, principal series representations, representations of GL(2,R). I will probably use some of this material in my course.

The document which pushed me from the state of "not having a clue what the trace formula was" to "actually having some understanding of what the trace formula was", was David Whitehouse's wonderful notes. Thank you David.

## Example sheets

There will be at least one, because it's here.

## Harder project

Due to the procedure involved in selecting who attends these summer schools, my understanding is that the background knowledge of the members of the audience will be very varied. For those that have ended up on the course and who know most of the material already, I have suggested a project that you can work on; it's about constructing, in a simple case, an "R eigenvariety" and a "T eigenvariety" in a situation where they are not expected to be isomorphic; more details are at this link. Here is Chris Birkbeck's masters thesis on the topic and here are my notes on the local arguments in Langlands' original paper.